A generalized framework for nodal first derivative summation-by-parts operators

Fernández, David C. Del Rey; Boom, Pieter D.; Zingg, David W.

doi:10.1016/j.jcp.2014.01.038

Cited by 134 publications

(134 citation statements)

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“…While SBP methods have been extended in a number of ways, for example see [8][9][10][11], the majority of these developments have been limited to one-dimensional operators that are applied to multi-dimensional problems using tensor-product operators in computational space. An interesting exception is the work by Nordström et al [12], which presents a vertex-centered second-order-accurate finite-volume scheme with the SBP property on unstructured grids.…”

Section: Introductionmentioning

confidence: 99%

“…Building on the generalization in [9], we presented an SBP definition in [13] (see also [14]) that is suitable for arbitrary, bounded subdomains with piecewise smooth, orientable boundaries. For diagonal-norm 1 multi-dimensional SBP operators that are exact for polynomials of total degree p, it was shown that the norm and corresponding nodes define a strong cubature rule that is exact for polynomials of degree 2p − 1.…”

Section: Introductionmentioning

confidence: 99%

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Simultaneous Approximation Terms for Multidimensional Summation-by-parts Operators

Hicken

Fernández

Zingg

2016

46th AIAA Fluid Dynamics Conference

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This paper is concerned with the accurate, conservative, and stable imposition of boundary conditions and inter-element coupling for multi-dimensional summation-byparts (SBP) finite-difference operators. More precisely, the focus is on diagonal-norm SBP operators that are not based on tensor products and are applicable to unstructured grids composed of arbitrary elements. We show how penalty terms -simultaneous approximation terms (SATs) -can be adapted to discretizations based on multi-dimensional SBP operators to enforce boundary and interface conditions. A general SAT framework is presented that leads to conservative and stable discretizations of the variable-coefficient advection equation. This framework includes the case where there are no nodes on the boundary of the SBP element at which to apply penalties directly. This is an important generalization, because elements analogous to Legendre-Gauss collocation, i.e. without boundary nodes, typically have higher accuracy for the same number of degrees of freedom. Symmetric and upwind examples of the general SAT framework are created using a decomposition of the symmetric part of an SBP operator; these particular SATs enable the pointwise imposition of boundary and inter-element conditions. We illustrate the proposed SATs using triangular-element SBP operators with and without nodes that lie on the boundary. The accuracy, conservation, and stability properties of the resulting SBP-SAT discretizations are verified using linear advection problems with spatially varying divergence-free velocity fields.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Simultaneous Approximation Terms for Multidimensional Summation-by-parts Operators

Hicken

Fernández

Zingg

2016

46th AIAA Fluid Dynamics Conference

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“…Significant error reductions have been observed for several model problems using such operators [17]. Further, operators defined on grids that do not match with the physical domain boundaries have been introduced in [4]. Even though experimental evidence suggest that accuracy results along the lines of Theorems 4 and 5 hold also in these cases, there is a lack of formal theory to support such claims.…”

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confidence: 99%

“…It follows from the accuracy conditions on D that the norm P defines a high order quadrature rule. To see this, we consider the so called compatibility conditions [14,4,16,12]: We multiply the first accuracy condition in Definition 2 from the left by (x i ) T P to obtain…”

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confidence: 99%

On the order of Accuracy of Finite Difference Operators on Diagonal Norm Based Summation-by-Parts Form

Linders¹,

Lundquist²,

Nordström³

2018

SIAM J. Numer. Anal.

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Abstract. In this paper we generalise results regarding the order of accuracy of finite difference operators on Summation-By-Parts (SBP) form, previously known to hold on uniform grids, to grids with arbitrary point distributions near domain boundaries. We give a definite proof that the order of accuracy in the interior of a diagonal norm based SBP operator must be at least twice that of the boundary stencil, irrespective of the grid point distribution near the boundary. Additionally, we prove that if the order of accuracy in the interior is precisely twice that of the boundary, then the diagonal norm defines a quadrature rule of the same order as the interior stencil. Again, this result is independent of the grid point distribution near the domain boundaries.Key words. Finite difference schemes, summation-by-parts operators, numerical differentiation, quadrature rules, order of accuracy AMS subject classifications. 65M06, 65N061. Introduction. Summation-By-Parts (SBP) operators, applied in the discretisation of systems of partial differential equations have received considerable attention since they lead to provable energy stability [25], and recently, entropy stability [6] for well-posed problems. Finite difference stencils on SBP form were first introduced in [14,15] based on central difference schemes of order 2 and 4. Later, operators with minimal bandwidth using stencils of order 6 and 8 were developed in [22]. In [3,19] SBP operators of orders up to 8 for both first and second derivatives were presented.The SBP concept has been extended to methods outside the finite difference community. These include spectral collocation and spectral element methods [2,8,9] as well as correction procedures via reconstruction [21]. Further, the finite difference class of SBP operators has been enlarged to multidimensional operators similar to Galerkin methods [12] as well as to grid dependent stencils akin to element based methods [5]. In this paper we restrict our attention to fixed stencil finite difference schemes and do not consider these extended approaches further.Implicit to the definition of an SBP operator is the notion of a discrete norm. If this norm is represented by a diagonal matrix, the associated operator is referred to as a diagonal norm based SBP operator. To avoid stability issues on curvilinear grids [23], and in general for problems with variable coefficients [20], finite difference operators on SBP form are in practice usually based on a diagonal norm.The focus in this paper is on SBP operators consisting of a repeated central difference stencil in the interior, and one-sided stencils near boundaries and interfaces. We introduce the notation SBP(τ ,2s) to refer to such an operator that is of order τ near the boundary and of order 2s in the interior. For completely uniform grid distributions, the accuracy of SBP(τ ,2s) is known to be dictated by two main constraints; we will formalise these as Theorems 4 and 5 in the next section. The first one states that s ≥ τ , i.e. the order of accuracy of the int...

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“…the same order as for the schemes in the interior, and hence higher than at other time levels in the boundaries of the blocks being of order s+1. 2,8,13 It should also be noted that many other types of SBP operators exist, see 9 for a comprehensive review which also includes a generalized approach. It has also been shown that the resulting time integration schemes form a subset of implicit Runge-Kutta schemes.…”

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confidence: 99%